At the end of this paper, we prove two convergence theorems of iterative schemes which approximate a fixed point of a nonexpansive mapping. Firstly, we apply the main result Theorem 3.2 to this problem. We begin with the following lemmas.
Lemma 5.1 A nonexpansive mapping defined on a CAT(1) space is Δdemiclosed.
Proof Let S:X\to X be a nonexpansive mapping. Let \{{x}_{n}\} be a Δconvergent sequence in X with the Δlimit {x}_{\mathrm{\infty}}\in X and suppose that {lim}_{n\to \mathrm{\infty}}d({x}_{n},S{x}_{n})=0. We will prove that {x}_{\mathrm{\infty}}=S{x}_{\mathrm{\infty}}. If {x}_{\mathrm{\infty}}\ne S{x}_{\mathrm{\infty}}, then, by the uniqueness of the asymptotic center, we have that
\begin{array}{rl}\underset{n\to \mathrm{\infty}}{lim\hspace{0.17em}sup}d({x}_{n},{x}_{\mathrm{\infty}})& <\underset{n\to \mathrm{\infty}}{lim\hspace{0.17em}sup}d({x}_{n},S{x}_{\mathrm{\infty}})\\ \le \underset{n\to \mathrm{\infty}}{lim\hspace{0.17em}sup}(d({x}_{n},S{x}_{n})+d(S{x}_{n},S{x}_{\mathrm{\infty}}))\\ \le \underset{n\to \mathrm{\infty}}{lim\hspace{0.17em}sup}(d({x}_{n},S{x}_{n})+d({x}_{n},{x}_{\mathrm{\infty}}))\\ =\underset{n\to \mathrm{\infty}}{lim\hspace{0.17em}sup}d({x}_{n},{x}_{\mathrm{\infty}}),\end{array}
a contradiction. Hence, we have that S is Δdemiclosed. □
Lemma 5.2 Let X be a CAT(1) space such that d({v}^{\prime},{v}^{\u2033})+d({v}^{\u2033},v)+d(v,{v}^{\prime})<2\pi for every v,{v}^{\prime},{v}^{\u2033}\in X. Let S:X\to X be a nonexpansive mapping with a nonempty set of fixed points F(S). Then the mapping T:X\to X defined by
Tx=\frac{1}{2}x\oplus \frac{1}{2}Sx
for x\in X is a strongly quasinonexpansive and Δdemiclosed mapping such that F(T)=F(S).
Proof It is obvious that F(T)=F(S) by definition and, since both the identity mapping I and S are quasinonexpansive, for x\in X and p\in F(T)=F(S), we have that
cosd(Tx,p)\ge \frac{1}{2}cosd(x,p)+\frac{1}{2}cosd(Sx,p)\ge cosd(x,p).
Thus T is quasinonexpansive. Let \{{x}_{n}\} be a sequence in X such that {sup}_{n\in \mathbb{N}}d({x}_{n},p)<\pi /2, and suppose that {lim}_{n\to \mathrm{\infty}}(cosd({x}_{n},p)/cosd(T{x}_{n},p))=1. Then we have
\begin{array}{rl}cosd(T{x}_{n},p)sind({x}_{n},S{x}_{n})& \ge sin\frac{d({x}_{n},S{x}_{n})}{2}(cosd({x}_{n},p)+cosd(S{x}_{n},p))\\ \ge 2sin\frac{d({x}_{n},S{x}_{n})}{2}cosd({x}_{n},p)\end{array}
for every n\in \mathbb{N}. It follows that
cosd(T{x}_{n},p)cos\frac{d({x}_{n},S{x}_{n})}{2}\ge cosd({x}_{n},p)
and since {sup}_{n\in \mathbb{N}}d(T{x}_{n},p)\le {sup}_{n\in \mathbb{N}}d({x}_{n},p)<\pi /2, we have
\begin{array}{rl}1& \ge \underset{n\to \mathrm{\infty}}{lim\hspace{0.17em}sup}cosd({x}_{n},T{x}_{n})\\ \ge \underset{n\to \mathrm{\infty}}{lim\hspace{0.17em}inf}cosd({x}_{n},T{x}_{n})\\ =\underset{n\to \mathrm{\infty}}{lim\hspace{0.17em}inf}cos\frac{d({x}_{n},S{x}_{n})}{2}\\ \ge \underset{n\to \mathrm{\infty}}{lim}\frac{cosd({x}_{n},p)}{cosd(T{x}_{n},p)}\\ =1.\end{array}
It implies that {lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0 and hence T is strongly quasinonexpansive.
For the Δdemiclosedness of T, use Lemmas 4.5 and 5.1 with the fact that the identity mapping is also Δdemiclosed. □
Applying this lemma and the results in the previous section to Theorem 3.2, we obtain the following convergence theorem of an iterative scheme approximating a fixed point of a nonexpansive mapping.
Theorem 5.3 Let X be a complete CAT(1) space such that d(v,{v}^{\prime})<\pi /2 for every v,{v}^{\prime}\in X. Let S:X\to X be a nonexpansive mapping and suppose that F(S)\ne \mathrm{\varnothing}. Let \{{\alpha}_{n}\} be a real sequence in ]0,1[ such that {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0 and {\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}. For given points u,{x}_{0}\in X, let \{{x}_{n}\} be the sequence in X generated by
{x}_{n+1}={\alpha}_{n}u\oplus (1{\alpha}_{n})(\frac{1}{2}{x}_{n}\oplus \frac{1}{2}S{x}_{n})
for n\in \mathbb{N}. Suppose that one of the following conditions holds:

(a)
{sup}_{v,{v}^{\prime}\in X}d(v,{v}^{\prime})<\pi /2;

(b)
d(u,{P}_{F(S)}u)<\pi /4 and d(u,{P}_{F(S)}u)+d({x}_{0},{P}_{F(S)}u)<\pi /2;

(c)
{\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}^{2}=\mathrm{\infty}.
Then \{{x}_{n}\} converges to {P}_{F(S)}u.
The next convergence theorem of an iterative scheme on CAT(1) spaces was first proposed by Pia̧tek [6]. The theorem deals with the Halperntype iterative sequence. Although the result itself is correct, a part of the proof does not seem to be exact. Precisely, in the proof of the convergence theorem for the explicit iteration process, the author makes use of Xu’s lemma, Lemma 2.1 in this paper. However, the conditions required for this lemma are not verified. We attempt to prove the following theorem as a supplement of the aforementioned result, and moreover, we find another coefficient condition which guarantees convergence of the iterative scheme.
Before showing the result, we need the following lemma which is analogous to [[6], Lemma 3.3]. The assumption for the length of the edges of the triangle is improved.
Lemma 5.4 Let X be a CAT(1) space. For M\in \phantom{\rule{0.2em}{0ex}}]0,\pi [, let u,v,w\in X be such that d(u,v)\le M and d(u,w)\le M. For a given \alpha \in \phantom{\rule{0.2em}{0ex}}]0,1[, let {v}^{\prime}=\alpha u\oplus (1\alpha )v and {w}^{\prime}=\alpha u\oplus (1\alpha )w. If d(v,w)+d(w,u)+d(u,v)<2\pi and sin((1\alpha )M)\le sinM, then
d({v}^{\prime},{w}^{\prime})\le \frac{sin((1\alpha )M)}{sinM}d(v,w).
Proof Consider the comparison triangle \mathrm{\u25b3}(\overline{u},\overline{v},\overline{w}) of \mathrm{\u25b3}(u,v,w) on {\mathbb{S}}^{2} and let {\overline{v}}^{\prime} and {\overline{w}}^{\prime} be the comparison points of {v}^{\prime} and {w}^{\prime}, respectively. Let
U={d}_{{\mathbb{S}}^{2}}(\overline{v},\overline{w}),\phantom{\rule{2em}{0ex}}V={d}_{{\mathbb{S}}^{2}}(\overline{w},\overline{u}),\phantom{\rule{2em}{0ex}}W={d}_{{\mathbb{S}}^{2}}(\overline{u},\overline{v}),
and {U}^{\prime}={d}_{{\mathbb{S}}^{2}}({\overline{v}}^{\prime},{\overline{w}}^{\prime}). Then, letting {\alpha}^{\prime}=1\alpha, we have that
Since the functions {f}_{V}(t)=sintV/sintM, {f}_{W}(t)=sintW/sintM, and g(t)=(1costVcostW)/(sintVsintW) are all increasing on [0,1], it follows that
sinVsinW\frac{{sin}^{2}{\alpha}^{\prime}M}{{sin}^{2}M}sin{\alpha}^{\prime}Vsin{\alpha}^{\prime}W\ge 0,
and
(1cosVcosW)sin{\alpha}^{\prime}Vsin{\alpha}^{\prime}WsinVsinW(1cos{\alpha}^{\prime}Vcos{\alpha}^{\prime}W)\ge 0.
Therefore, we have that
sinVsinW(\frac{{sin}^{2}{\alpha}^{\prime}M}{{sin}^{2}M}(1cosU)(1cos{U}^{\prime}))\ge 0.
Using the assumption that sin{\alpha}^{\prime}M\le sinM, we obtain that
\begin{array}{rl}sin\frac{{U}^{\prime}}{2}& =\sqrt{\frac{1cos{U}^{\prime}}{2}}\le \frac{sin{\alpha}^{\prime}M}{sinM}\sqrt{\frac{1cosU}{2}}\\ =\frac{sin{\alpha}^{\prime}M}{sinM}sin\frac{U}{2}\le sin\left(\frac{sin{\alpha}^{\prime}M}{sinM}\frac{U}{2}\right)\end{array}
and, by the CAT(1) inequality, it follows that
d({v}^{\prime},{w}^{\prime})\le {d}_{{\mathbb{S}}^{2}}({\overline{v}}^{\prime},{\overline{w}}^{\prime})={U}^{\prime}\le \frac{sin((1\alpha )M)}{sinM}U=\frac{sin((1\alpha )M)}{sinM}d(v,w),
which is the desired result. □
Theorem 5.5 Let X be a complete CAT(1) space such that d(v,{v}^{\prime})<\pi /2 for every v,{v}^{\prime}\in X. Let T:X\to X be a nonexpansive mapping and suppose that F(T)\ne \mathrm{\varnothing}. Let \{{\alpha}_{n}\} be a real sequence in [0,1] such that {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0, {\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}, and {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n+1}{\alpha}_{n}<\mathrm{\infty}. For given points u,{x}_{0}\in X, let \{{x}_{n}\} be the sequence in X generated by
{x}_{n+1}={\alpha}_{n}u\oplus (1{\alpha}_{n})T{x}_{n}
for n\in \mathbb{N}. Suppose that one of the following conditions holds:

(a)
{sup}_{v,{v}^{\prime}\in X}d(v,{v}^{\prime})<\pi /2;

(b)
d(u,{P}_{F(T)}u)<\pi /4 and d(u,{P}_{F(T)}u)+d({x}_{0},{P}_{F(T)}u)<\pi /2;

(c)
{\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}^{2}=\mathrm{\infty}.
Then \{{x}_{n}\} converges to {P}_{F(T)}u.
We employ the method used in [6] for some parts of the proof.
Proof From the definition of \{{x}_{n}\}, using Lemma 5.4, we have that
\begin{array}{rl}d({x}_{n},{x}_{n+1})& \le d({x}_{n},{\alpha}_{n}u\oplus (1{\alpha}_{n})T{x}_{n1})+d({\alpha}_{n}u\oplus (1{\alpha}_{n})T{x}_{n1},{x}_{n+1})\\ ={\alpha}_{n}{\alpha}_{n1}d(u,T{x}_{n1})+d({\alpha}_{n}u\oplus (1{\alpha}_{n})T{x}_{n1},{x}_{n+1})\\ \le {\alpha}_{n}{\alpha}_{n1}d(u,T{x}_{n1})+\frac{sin((1{\alpha}_{n}){M}_{n})}{sin{M}_{n}}d(T{x}_{n1},T{x}_{n})\\ \le {\alpha}_{n}{\alpha}_{n1}d(u,T{x}_{n1})+\frac{sin((1{\alpha}_{n}){M}_{n})}{sin{M}_{n}}d({x}_{n1},{x}_{n}),\end{array}
where {M}_{n}=max\{d(u,T{x}_{n}),d(u,T{x}_{n1})\} for each n\in \mathbb{N}. Let
{\gamma}_{n}=\{\begin{array}{cc}1\frac{sin((1{\alpha}_{n}){M}_{n})}{sin{M}_{n}}\hfill & ({M}_{n}\ne 0),\hfill \\ {\alpha}_{n}\hfill & ({M}_{n}=0)\hfill \end{array}
for all n\in \mathbb{N}. Then, as in the proof of Theorem 3.2, we have that each of the conditions (a), (b) and (c) implies that {\sum}_{n=0}^{\mathrm{\infty}}{\gamma}_{n}=\mathrm{\infty}. In particular, in cases of (a) and (b), for M={sup}_{v,{v}^{\prime}\in X}d(v,{v}^{\prime}) and M=max\{2d(u,p),d(u,p)+d({x}_{0},p)\} with p={P}_{F(T)}u, respectively, it holds that
{\gamma}_{n}\ge {\alpha}_{n}cos{M}_{n}\ge {\alpha}_{n}cosM,
and in case of (c), it holds that
{\gamma}_{n}\ge \frac{{\alpha}_{n}^{2}{\pi}^{2}}{8}.
Then, by using Lemma 2.1 with the condition {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n+1}{\alpha}_{n}<\mathrm{\infty}, we have that {lim}_{n\to \mathrm{\infty}}d({x}_{n},{x}_{n+1})=0. It follows that
0\le d({x}_{n},T{x}_{n})\le d({x}_{n},{x}_{n+1})+d({x}_{n+1},T{x}_{n})=d({x}_{n},{x}_{n+1})+{\alpha}_{n}d(u,T{x}_{n})
and thus {lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0. Let p, \{{s}_{n}\}, \{{t}_{n}\}, \{{\beta}_{n}\} be as in the proof of Theorem 3.2 again. Then by Lemma 3.1, we have that
{s}_{n+1}\le (1{\beta}_{n}){s}_{n}+{\beta}_{n}{t}_{n}
for every n\in \mathbb{N}. Since every nonexpansive mapping is Δdemiclosed, we can use the same technique as the proof of Theorem 3.2 and then we obtain that
\begin{array}{rl}\underset{n\to \mathrm{\infty}}{lim\hspace{0.17em}sup}{t}_{n}& =\underset{n\to \mathrm{\infty}}{lim\hspace{0.17em}sup}(1\frac{cosd(u,p)}{sind(u,T{x}_{n})tan(\frac{{\alpha}_{n}}{2}d(u,T{x}_{n}))+cosd(u,T{x}_{n})})\\ \le 0.\end{array}
Consequently, we have that {lim}_{n\to \mathrm{\infty}}{s}_{n}=0 by Lemma 2.1. Hence, \{{x}_{n}\} converges to p={P}_{F(T)}u, which is the desired result. □